Daniele Loiacono

Graph Mining and Social Network Analysis Data Mining and Text Mining (UIC 583 @ Politecnico di Milano)

Daniele Loiacono

References

q  Jiawei Han and Micheline Kamber, "Data Mining: Concepts and Techniques", The Morgan Kaufmann Series in Data Management Systems (Second Edition) " Chapter 9

Graph Mining

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Graph Mining Overview

q  Graphs are becoming increasingly important to model many phenomena in a large class of domains (e.g., bioinformatics, computer vision, social analysis)

q  To deal with these needs, many data mining approaches have been extended also to graphs and trees

q  Major approaches " Mining frequent subgraphs " Indexing " Similarity search " Classification " Clustering

Daniele Loiacono

Mining frequent subgraphs

q  Give n a labeled graph data set

D = {G1,G2,…,Gn}

q  We define support(g) as the percentage of graphs in D where g is a subgraph

q  A frequent subgraph in D is a subgraph with a support greater than min_sup

q  How to find frequent subgraph? " Apriori-based approach " Pattern-growth approach

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AprioriGraph

q  Apply a level-wise iterative algorithm 1.  Choose two similar size-k frequent subgraphs in S 2.  Merge two similar subgraphs in a size-(k+1) subgraph 3.  If the new subgraph is frequent add to S 4.  Restart from 2. until all similar subgraphs have been

considered. Otherwise restart from 1. and move to k+1. q  What is subgraph size?

" Number of vertex " Number of edges " Number of edge-disjoint paths

q  Two subgraphs of size-k are similar if they have the same size-(k-1) subgraph

q  AprioriGraph has a big computational cost (due to the merging step)

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PatternGrowthGraph

q  Incrementally extend frequent subgraphs 1.  Add to S each frequent subgraphs gE obtained by

extending subgraph g 2.  Until S is not empty, select a new subgraph g in S to

extend and start from 1. q  How to extend a subgraph?

" Add a vertex " Add an edge

q  The same graph can be discovered many times! " Get rid of duplicates once discovered " Reduce the generation of duplicates

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Mining closed, unlabeled, and constrained subgraphs

q  Closed subgraphs " G is closed iff there is no proper supergraph G’ with the

same support of G " Reduce the growth of subgraphs discovered " Is a more compact representation of knowledge

q  Unlabeled (or partially labeled) graphs " Introduce a special label Φ " Φ can match any label or only itself

q  Constrained subgraphs " Containment constraint (edges, vertex, subgraphs) " Geometric constraint " Value constraint

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Graph Indexing

q  Indexing is basilar for effective search and query processing q  How to index graphs? q  Path-based approach takes the path as indexing unit

" All the path up to maxL length are indexed " Does not scale very well

q  gIndex approach takes frequent and discriminative subgraphs as indexing unit " A subgraph is frequent if it has a support greater than a

threshold " A subgraph is discriminative if its support cannot be well

approximated by the intersection of the graph sets that contain one of its subgraphs

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Graph Classification and Clustering

q  Mining of frequent subgraphs can be effectively used for classification and clustering purposes

q  Classification " Frequent and discriminative subgraphs are used as features to

perform the classification task " A subgraph is discriminative if it is frequent only in one class of

graphs and infrequent in the others " The threshold on frequency and discriminativeness should be

tuned to obtain the desired classification results q  Clustering

" The mined frequent subgraphs are used to define similarity between graphs

" Two graphs that share a large set of patterns should be considered similar and grouped in the same cluster

" The threshold on frequency can be tuned to find the desired number of clusters

q  As the mining step affects heavily the final outcome, this is an intertwined process rather tan a two-steps process

Social Network Analysis

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Social Network

q  A social network is an heterogeneous and multirelational dataset represented by a graph " Vertexes represent the objects (entities) " Edges represent the links (relationships or interaction) " Both objects and links may have attributes " Social networks are usually very large

q  Social network can be used to represents many real-world phenomena (not necessarily social) " Electrical power grids " Phone calls " Spread of computer virus " WWW

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Small World Networks (1)

q  Are social networks random graphs? q  NO!

Internet Map Science

Coauthorship Protein Network

Few degrees of separation

High degree of local clustering

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Peter Jane

Sarah

Ralph Society: Six degrees S. Milgram 1967 F. Karinthy 1929 WWW: 19 degrees Albert et al. 1999

Small World Networks (2)

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Small World Networks (3)

q  Definitions " Node’s degree is the number of incident edges " Network effective diameter is the max distance within 90% of

the network

q  Properties " Densification power law

" Shrinking diameter " Heavy-tailed degrees distribution

n: number of nodes e: number of eges 1<α<2

Node rank

Nod

e deg

rees

Preferential attachment model

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Mining social networks (1)

q  Several Link mining tasks can be identified in the analysis of social networks

q  Link based object classification " Classification of objects on the basis of its attributes, its

links and attributes of objects linked to it " E.g., predict topic of a paper on the basis of

•  Keywords occurrence •  Citations and cocitations

q  Link type prediction " Prediction of link type on the basis of objects attributes " E.g., predict if a link between two Web pages is an

advertising link or not q  Predicting link existence

" Predict the presence of a link between two objects

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Mining social networks (2)

q  Link cardinality estimation " Prediction of the number of links to an object " Prediction of the number of objects reachable from a

specific object q  Object reconciliation

" Discover if two objects are the same on the basis of their attributes and links

" E.g., predict if two websites are mirrors of each other q  Group detection

" Clustering of objects on the basis both of their attributes and their links

q  Subgraph detection " Discover characteristic subgraphs within network

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Challenges

q  Feature construction " Not only the objects attributes need to be considered but

also attributes of linked objects " Feature selection and aggregation techniques must be

applied to reduce the size of search space q  Collective classification and consolidation

" Unlabeled data cannot be classified independently " New objects can be correlated and need to be considered

collectively to consolidate the current model q  Link prediction

" The prior probability of link between two objects may be very low

q  Community mining from multirelational networks " Many approaches assume an homogenous relationship

while social networks usually represent different communities and functionalities

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Applications

q  Link Prediction q  Viral Marketing

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Link prediction

q  What edges will be added to the network? q  Given a snapshot of a network at time t, link prediction aims

to predict the edges that will be added before a given future time t’

q  Link prediction is generally solved assigning to each pair of nodes a weight score(X,Y)

q  The higher the score the more likely that link will be added in the near future

q  The score(X,Y) can be computed in several way " Shortest path: the shortest he path between X and Y the

highest is their score " Common neighbors: the greater the number of neighbors

X and Y have in common, the highest is their score " Ensemble of all paths: weighted sum of paths that

connects X and Y (shorter paths have usually larger weights)

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Viral Marketing

q  Several marketing approaches " Mass marketing is targeted on specific segment of

customers " Direct marketing is target on specific customers solely on

the basis of their characteristics " Viral marketing tries to exploit the social connections to

maximize the output of marketing actions q  Each customer has a specific network value based on

" The number of connections " Its role in the network (e.g., opinion leader, listener) " Role of its connections

q  Viral marketing aims to exploit the network value of customers to predict their influence and to maximize the outcome of marketing actions

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Viral Marketing: Random Spreading

q  500 randomly chosen customers are given a product (from 5000).

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Viral Marketing: Directed Spreading

q  The 500 most connected consumers are given a product.